# Darboux Integrability of Trapezoidal $H^{4}$ and $H^{6}$ Families of Lattice Equations II: General Solutions

## TL;DR

This paper derives general solutions for specific lattice equations using Darboux integrability and first integrals, reducing the problem to solvable linear or linearizable difference equations.

## Contribution

It provides explicit general solutions for trapezoidal H^4 and H^6 lattice equations by leveraging Darboux integrability properties.

## Key findings

- Solutions expressed via linear or linearizable difference equations.
- Extension of Darboux integrability methods to new lattice equations.
- Formal solutions obtained for complex quad-equations.

## Abstract

In this paper we construct the general solutions of two families of quad-equations, namely the trapezoidal $H^{4}$ equations and the $H^{6}$ equations. These solutions are obtained exploiting the properties of the first integrals in the Darboux sense, which were derived in [Gubbiotti G., Yamilov R.I., J. Phys. A: Math. Theor. 50 (2017), 345205, 26 pages, arXiv:1608.03506]. These first integrals are used to reduce the problem to the solution of some linear or linearizable non-autonomous ordinary difference equations which can be formally solved.